The Finite Difference Methods for Parabolic Partial Differential Equations
Received Date: Aug 18, 2018 / Accepted Date: Sep 12, 2018 / Published Date: Sep 27, 2018
Keywords: Parabolic; Heat equation; Finite difference; Bender- Schmidt; Crank-Nicolson
Parabolic partial differential equations
The well-known parabolic partial differential equation is the one dimensional heat conduction equation . The solution of this equation is a function u(x,t) which is defined for values of x from 0 to l and for values of t from 0 to ∞ [2-4]. The solution is not defined in a closed domain but advances in an open ended region from initial values, satisfying the prescribed boundary conditions (Figure 1). In general the study of pressure waves in a fluid, propagation of heat and unsteady state problems lead to parabolic type of equations .
Solution of one-dimensional Heat conduction equation
Consider the heat conduction which is of the form 
where C is a constant Let the plane (x,t) be divided in to smaller rectangles by means of the sets of lines 
Hence the finite difference analogue in eqn. (1) is
Which can be rewritten as
where this formula expresses the unknown function value at the (i,j+1)th interior point in terms of the known function values and hence it is called the explicit formula. It can be shown that this formula is valid only  for For becomes in eqn. (4)
The above formula in eqn. (5) is called Bender-Schmidt recurrence relation. In formula in eqn. (4), we have used the function values along the jth row only in the approximation of uxx is replaced by the  average of its finite difference approximations on the jth and (i,j+1)th rows. Thus
and hence in eqn. (1) is replaced by
where , on the left side in eqn. (6) we have three unknowns and on the right side all the three quantities are known. The above formula in eqn. (6) which is an implicit scheme is called Crank-Nicolson formula and it is convergent for all finite values of λ. If there are N internal mesh points on each row, then the formula in eqn. (6) gives N simultaneous equations for the N unknowns in terms of the given boundary values. Similarly the internal mesh points on all rows can be calculated .
Example 1: Use the Bender-Schmidt recurrence relation to solve the equation uxx=2ut. With the conditions u(x,0)=4x−x2,u(0,t)=u(4,t)=0 Solution Taking h=1 we obtain 
Also u(0,0)=0,u(1,0)=3,u(2,0)=4,u(3,0)=3 and u(4,0)=0. For this the first time step, k=1.
For this the first time step, k=1.
Using Bender-Schmidt recurrence relation, we obtain
For k=2, we have
For k=3 we have
For k=4 we have
Similarly with k=5 we obtain
The computation can be continued to any number of time steps.
Example 2: Solve the equation ut=uxx, subject to the conditions,
u(x,0)=sinπx,0 ≤ x ≤ 1,u(0,t)=u(1,t)=0.
(a) Bender-Schmidt Method
(b) Crank-Nicolson Method
Carry out the computations for two levels, taking
Here and so that
Also and all boundary values are zero in Figure 2.
(a) Using Bender-Schmidt formula
Using in eqn. (4) of this section
we have ui,j+1=λui−1,j+(1−2λ)ui,j+λui+1,j
For i=1,2; j=0
(b) By using Crank-Nicolson method
From eqn. (6) above we have
For i=1,2; j=0
Solving these equations, we find
For i=1,2; j=1
−u0,2+10u1,2−u2,2=u0,1+ 6u1,1+ u2,1
Again solving the above two equation we can obtain
Solution of two dimensional heat equations
The method employed for the solution of one dimensional heat equation can be readily extended to the solution two dimensional heat equations in eqn. (7).
Consider a square region 0 ≤ x ≤ y ≤ a and assume that u is known at all points within and on the boundary of this square.
If h be the step size then a mesh point
(x,y,t) =(ih,jh,nl) may be denoted as (i,j,n).
Replacing the derivatives in eqn. (7) by their finite difference approximations, we get
Where . This equation needs the five points available on the nth plane (Figure 3).
The computation process consists point by point evaluation in the (n+1)th plane using the points on the nth plane. This method is known as ADE (Alternating Direct Explicit) method.
Example: Solve the equation ut=uxx+uyy.
Subject to the initial conditions u(x,y,0) =sin2πxsin2πy,0 ≤ x,y ≤ 1 and the conditions u(x,y,t)=0, t ≥ 0, on the boundaries using ADE method and (Calculate the results for one time level).
Solution: In eqn. (8) becomes
The mesh points and the computation model is given in Figure 4 below
At the zeroth level (n=0), the initial and boundary conditions are
Now we can calculate the mesh values at the first level.
For n=0, from (∗) we obtain
(i) Putting i=j=1 in (∗∗) above
(ii) Putting i=2,j=1 in(∗∗) above
(iii) Putting i=1,j=2 in (∗∗) above
(iv) Putting i=j=2 in (∗∗) above
Similarly the mesh values at the second and higher levels can be calculated.
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Citation: Nigatie Y (2018) The Finite Difference Methods for Parabolic Partial Differential Equations. J Appl Computat Math 7: 418. DOI: 10.4172/21689679.1000418
Copyright: © 2018 Nigatie Y. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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